On split metacyclic groups
Using the properties of primitive characters, Gauss sums, and the Ramanujan sum, we study two hybrid mean values of Gauss sums and generalized Bernoulli numbers and give two asymptotic formulas.
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| author | Liu, He-guo Yan, Yang Лю, Хе-го Ян, Янг |
| author_facet | Liu, He-guo Yan, Yang Лю, Хе-го Ян, Янг |
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| description | Using the properties of primitive characters, Gauss sums, and the Ramanujan sum, we study two hybrid mean values of Gauss sums and generalized Bernoulli numbers and give two asymptotic formulas. |
| first_indexed | 2026-03-24T02:27:59Z |
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UDC 512.5
Yan Yang, He-guo Liu (Hubei Univ., Wuhan; Hubei Univ. Arts and Sci., Xiangyang, China)
ON SPLIT METACYCLIC GROUPS *
ПРО РОЗЩЕПЛЮВАНI МЕТАЦИКЛIЧНI ГРУПИ
We consider sufficient conditions for metacyclic groups to split. Speciifically, we show that a finite metacyclic group of
odd order G is split on its cyclic normal subgroup K if K is such that G/K is cyclic and |K| = expG.
Розглянуто достатнi умови для розщеплення метациклiчних груп, а саме, показано, що скiнченна метациклiчна
група G непарного порядку розщеплюється на своїй циклiчнiй нормальнiй пiдгрупi K, якщо K є такою, що G/K
є циклiчною та |K| = expG.
1. Introduction. G is a metacyclic group if and only if there exists a cyclic normal subgroup K of
G such that G/K is cyclic, and G = SK is called a metacyclic factorization, when S is cyclic. In
particular, if G has a split metacyclic factorization G = SK such that S ∩K = 1, then G is called
split metacyclic group, otherwise is called nonsplit. C. E. Hempel and Hyo-Seob Sim have given the
classifications of metacyclic groups in their papers [1, 2].
Resently, the structure of the automorphism group of metacyclic group have been given much
attention. The automorphism groups of the split metacyclic p-groups have been given in [3] for p is
odd and in [4] for p = 2. And the automorphism groups of the finite split metacyclic groups have
been given in [5]. But the case of nonsplit groups is much more complicated than the case of split
group, and only the automorphism groups of nonsplit metacyclic p-group of odd order have been
given in [6]. Thus it is necessary to determine wether the metacyclic group is split or not, before we
study its automorphism. And in this paper, of particular interest are some sufficient coditions to show
a special type of finite metacyclic groups are split.
2. Notation and preliminaries. In this section, we present some general facts that will be useful
in this paper. We first define some notation which will be kept throughout.
π(G): the set of all prime divisors of the order of a finite group G.
r(p): the largest integer i such that pi divides the positive integer r.
G is a metacyclic group if and only if there exists a cyclic normal subgroup K of G such that
G/K is cyclic. And such a subgroup K is called a kernel of G.
Lemma 1. A p-group P of odd order is metacyclic with a kernel of order pγ and of index pα
if and only if it has a presentation
P = 〈a, b | apα = bp
β
, bp
γ
= 1, ba = bp
δ+1〉,
where α, β, γ, δ are positive integers such that γ ≤ min(α+ δ, β + δ).
Lemma 2 ([1], Lemma 2.1). A group G is metacyclic with a kernel of order γ and of index α if
and only if it has a presentation
G = 〈a, b | aα = bβ, bγ = 1, ba = bδ〉,
where α, β, γ, δ are positive integers such that γ|δα − 1 and γ|β(δ − 1).
*This work was supported by the Natural Science Foundation of Hubei Univ. Arts and Sci. (Grant No.2010YA002).
c© YAN YANG, HE-GUO LIU, 2012
1432 ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 10
ON SPLIT METACYCLIC GROUPS 1433
Lemma 3 ([2], Lemma 3.4). Let P = SK be a metacyclic factorization of a p-group P of odd
order. P is present by the presentation
P = 〈x, y |xpα = yp
β
, yp
γ
= 1, yx = yp
δ+1〉,
where
pα = |S : S ∩K|, pβ = |K : S ∩K|, pγ = |K|, pδ = |K : P ′|.
Lemma 4 ([5], Corollary 4.1). Let p be a odd prime, k, s, t be non-negative integers. Then
(i) (1 + pk)p
st ≡ 1 (mod pk+s);
(ii) if k > 0, then
1 + rs + . . .+ rs(p
t−1) ≡ pt (mod pt+k),
for r = 1 + pk.
Lemma 5 ([2], Lemma 5.3). Let G be a metacyclic group with a metacyclic factorization G =
= SK. To each set π of prime numbers, the subgroup H = SπKπ is the unique Hall π-subgroup of
G such that Sπ = S ∩H and Kπ = K ∩H, so H = (S ∩H)(K ∩H).
Definition 1 ([2], Definition 5.4). Let G be a group with a metacyclic factorization G = SK
and let π denote the set {p ∈ π(G) : G has a normal Hall p′-subgroup}. Let H denote the Hall
π-subgroup SπKπ and let N denote the the Hall π′-subgroup Sπ′Kπ′ . Then the semidirect decompo-
sition G = HN is called the standard Hall-decomposition for the metacyclic factorization G = SK.
Lemma 6 ([2], Lemma 5.6). Let G = HN be a Hall-decomposition for the metacyclic factor-
ization G = SK and π={p ∈ π(G) : G has a normal Hall p′-subgroup}. Then
(i) H = SπKπ = (S ∩H)(K ∩H), and H is nilpotent;
(ii) N = Sπ′Kπ′ = (S ∩N)(K ∩N), and Kπ′ = G′ ∩N, Sπ′ ∩Kπ′ = 1;
(iii) Sπ′ ≤ CN (H).
3. Split metacyclic group. 3.1. Split metacyclic p-group.
Lemma 7. If P = SK be a metacyclic factorization of a p-group P of odd order, then
expP = max (|S|, |K|).
Proof. By Lemma 3, P can be represented as
P = 〈a, b|apα = bp
β
, bp
γ
= 1, ba = bp
δ+1〉,
where |a| = |S| and |b| = |K|. For any asbt ∈ P, by Lemma 4, we have
(asbt)p
m
= asp
m
bt(1+(pδ+1)s+...+(pδ+1)s(p
m−1)) = asp
m
bt(p
m+kpδ+m) = 1,
where pm = max (|S|, |K|) and k is an integer.
Lemma 8. P is a metacyclic p-group of odd order. If K is a normal cyclic subgroup with
|K| = expP, then P/K is cyclic.
Proof. Let
P = 〈a, b|apα = bp
β
, bp
γ
= 1, ba = bp
δ+1〉,
be a presentation of P. Suppose K = 〈ambn〉, and we observe that |K| = expP = max(|a|, |b|)
from Lemma 7. Then by Lemma 4 we have
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 10
1434 YAN YANG, HE-GUO LIU
(apbp)p
γ−1
= ap
γ
bp(1+(1+pδ)p+...+(1+pδ)p(p
γ−1−1)) = 1,
which shows (m, p) = 1 or (n, p) = 1. If (m, p) = 1, we know that P/K is cyclic since P =
= 〈b〉〈ambn〉. If (n, p) = 1, P/K is also cyclic since P = 〈a〉〈ambn〉.
Theorem 1. Let P be a metacyclic p-group of odd order. If P has a normal cyclic subgroup K
of order expP, then P is split.
Proof. P/K is cyclic by Lemma 8. Let P = HK be a metacyclic factorization of P. Thus by
Lemma 3, choosing the generators x and y for H and K, respectively, we have the presentation
P = 〈x, y |xpα = yp
β
, yp
γ
= 1, yx = yp
δ+1〉,
where
pα = |H : H ∩K| = |P |/ expP, pβ = |K : H ∩K|,
pγ = |K| = expP, pδ = |K : P ′| = |P/P ′|/pα.
If β = γ, then P is split. Now we assume that β < γ. Denote a = xyfp
β−α
, where f = pγ−β − 1,
then we will show 〈a〉 ∩ 〈y〉 = 1. From Lemma 1 we can obtain
ap
α
= (xyfp
β−α
)p
α
= xp
α
yfp
β−α(1+(pδ+1)+...+(pδ+1)p
α−1) = xp
α
yfp
β−αpα =
= yp
β+fpβ = yp
γ
= 1,
and
ap
α−1
= (xyfp
β−α
)p
α−1
= xp
α−1
yfp
β−αpα−1
/∈ K,
which implies 〈a〉 ∩ 〈y〉 = 1. P = 〈a〉〈y〉 is obvious. Thus P is split. Finally, because of
ya = yxy
fpβ−α
= yx = y1+p
δ
,
we can get the presentation of P
P = 〈a, y | apα = yp
γ
= 1, ya = yp
δ+1〉,
where pα = |P |/ expP, pγ = expP, pα+δ = |P/P ′|.
Example 1. The above theorem may not hold for a metacyclic 2-group
Q8 = 〈x, a |x2 = a2, a4 = 1, ax = a−1〉,
where |a| = 4 = expQ8 and 〈a〉 C Q8, but Q8 is not a split group.
Applying Theorem 1 and Lemma 1, we can deduce the following corollary.
Corollary 1. A metacyclic p-group P of odd order is non-split if and only if P has the presen-
tation of the form in Lemma 1
P = 〈a, b | apα = bp
β
, bp
γ
= 1, ba = bp
δ+1〉,
where α > β > δ ≥ 1 and β < γ ≤ β + δ.
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 10
ON SPLIT METACYCLIC GROUPS 1435
3.2. Split metacyclic group. In the remaining part of this section, we will proof that the conclusion
above is also hold for a metacyclic group of odd order.
Lemma 9. Let P be a non-abelian metacyclic p-group of odd order. Suppose P = SK and
P = S1K1 both are the split metacyclic factorizations. Then
(i) |S| = |S1| and |K| = |K1|;
(ii) P = SK1 and P = S1K also are the split metacyclic factorizations. Here, P = SK is a
split metacyclic factorization means P = SK is a metacyclic factorization and S ∩K = 1.
Proof. (i) Suppose
P = SK = 〈a, b | apα = bp
γ
= 1, ba = bp
δ+1〉,
P = S1K1 = 〈a1, b1 | ap
α1
1 = bp
γ1
1 = 1, ba11 = bp
δ1+1
1 〉.
It is obvious that |P | = pα+γ = pα1+γ1 and expP = max (α, γ) = max (α1, γ1). Thus if α = α1,
(i) holds by Lemma 7. If α 6= α1, then α = γ1 and α1 = γ. And we observe that P/P ′ ∼=
∼= Zpα × Zpδ ∼= Zpα1 × Zpδ1 . Consequently,
γ1 = α = δ1, γ = α1 = δ,
which contradicts that P is non-abelian. Thus α = α1, γ = γ1, δ = δ1.
(ii) Let b1 = ambn, where m,n are non-negative integers. We will show that (n, p) = 1. Suppose
p|n. Then
bp
δ
1 = (ambn)p
δ
= bp(1+(1+pδ)m+...+(1+pδ)m(pδ−1)
) ∈ 〈bpδ+1〉,
which contradicts that P ′ = 〈bpδ〉 = 〈bp
δ
1 〉 = 〈(ambn)p
δ〉, since |K1/P
′| = |K/P ′| = pδ. Therefore,
P = 〈a, b〉 = 〈a, b1〉 = SK1 and S ∩K1 = 1.
Theorem 2. Let G be a metacyclic group of odd order and Y is a normal cyclic subgroup of
G with |Y | = expG. Then G/Y is cyclic and G is split .
Proof. Let G = HN be a Hall-decomposition for a metacyclic factorization G = SK, π =
= π(H)={p ∈ π(G) : G has a normal Hall p′-subgroup} and π′ = π(N). Thus Sπ = S ∩ H,
Kπ = K ∩H,Sπ′ = S ∩N, Kπ′ = K ∩N, by Lemma 5.
For p ∈ π, denote Hp and Yp = 〈yp〉 as the sylow p-subgroup of H and Y, respectively. Thus
Yp � Hp and |Yp| = expHp. Then from Lemma 9, we know Hp is split. Applying Lemma 9, we
can find a xp ∈ Hp, such that Hp = 〈xp〉〈yp〉 is a split metacyclic factorization. Let x1 =
∏
p∈π
xp,
y1 =
∏
p∈π
yp. Then H = 〈x1〉〈y1〉 is a split metacyclic factorization, since H is nilpotent.
If N = 1, then G = H is split. If N is non-trivial, then from Lemma 7, we know N = Sπ′Kπ′
is a split metacyclic factorazition, since Sπ′ ∩ Kπ′ = 1. Next we will show N = Sπ′Yπ′ is a split
metacyclic factorization, where Yπ′ is the Hall π′-subgroup of Y.
For any q ∈ π′, denote Yq and Nq = SqKq as the the sylow q-subgroup of Y and N, respectively.
Note that Yq � Nq and |Yq| = expNq. We know Yq is a kernel of Nq by Theorem 1. This implies
that Nq = SqYq is a split metacyclic factorization by Sq ∩Kq = 1 and Lemma 9. Thus
N =
∏
q∈π(N)
Nq =
∏
q∈π(N)
SqKq =
∏
q∈π(N)
SqYq =
∏
q∈π(N)
Sq
∏
q∈π(N)
Yq = Sπ′Yπ′ .
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 10
1436 YAN YANG, HE-GUO LIU
Futher, Sπ′ ∩ Yπ′ = 1, since Sq ∩ Yq = 1.
Let Sπ′ = 〈x2〉 and Yπ′ = 〈y2〉, then Sπ′ ≤ CN (H) and Y �G yields that
G = HN = 〈x1〉〈y1〉〈x2〉〈y2〉 = 〈x1〉〈x2〉〈y1〉〈y2〉 = 〈x1x2〉〈y1y2〉 = XY,
where X = 〈x1x2〉 and X ∩ Y = 1.
The above theorem may not hold for a group of even order.
Example 2. Let
G ∼= Q8 × Z3 = 〈x, b |x2 = b6, b12 = 1, bx = b7〉,
where |b| = 12 = expG and 〈b〉�G, but G is not a split group.
In Theorem 2, if Y is not normal, G may not be a split group.
Example 3. Let
G = 〈a, b | a34 = b3
3
, b3
5
= 1, ba = b1+32〉.
It is obviously that |a| = expG, but
[b, a] = b−1ba = b3
2 ∈ 〈a〉
shows 〈a〉 is not normal. And G is non-split.
Lemma 10. If G = SK is a metacyclic factorization of a metacyclic group G of odd order.
Then expG = lcm (|S|, |K|).
Proof. Let S = 〈x〉,K = 〈y〉, we will show that
|xmyn| ≤ lcm (|S|, |K|) ∀xmyn ∈ G.
Let r = lcm(|S|, |K|) and t = |K|. Suppose yx = yθ, and it is obvious that θr ≡ 1(mod |K|).
Then for p ∈ π(G) and kpr(p) = r
θr = θkp
r(p) ≡ 1 (mod pt(p)),
which implies θmkp
r(p) ≡ 1(mod pt(p)). Then from Lemma 4 we have
1 + θmk + . . .+ θmk(p
r(p)−1) ≡ pr(p)(mod pt(p)).
Thus
1 + θm + . . . θm(r−1) =
k−1∑
i=0
θmi(1 + θmk + . . .+ θmk(p
r(p)−1)) =⇒
=⇒ pt(p)|1 + θm + . . .+ θm(r−1) =⇒ t|1 + θm + . . .+ θm(r−1) =⇒
=⇒ (xmyn)r = xmryn(1+θ
m+...+θm(r−1)) = 1.
Corollary 2. Let G = SK be a metacyclic factorization of a metacyclic group G of odd order,
and |K|
∣∣|S|. Then G is split.
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 10
ON SPLIT METACYCLIC GROUPS 1437
Example 4. Let G = SK = 〈a〉〈b〉 be a metacyclic factorization.
(i) Suppose G = 〈a, b | a32 = b3
3
, b3
5
= 1, ba = b1+33〉. Then |K| = 35 = expG yields that G is
split. Let x = ab24, and we can get a presentation of G
G = 〈x, b |x32 = 1, b3
5
= 1, ba = b1+33〉.
(ii) Suppose G = 〈a, b | a34 = b3
3
, b3
5
= 1, ba = b1+33〉. Then S � G and |S| = 36 = expG
yields that G is split, for P ′ = 〈b33〉� S. Let x = b−1a6, and we can get a presentation of G
G = 〈x, a |x33 = 1, a3
6
= 1, ax = b1+34〉.
(iii) Suppose G = 〈a, b | a33 = b3
3×7, b3
4×7 = 1, ba = b415〉. Then |K| = 34 × 7 = expG yields
that G is split. Let x = ab14, and we can get a presentation of G
G = 〈x, b |x33 = 1, b3
4×7 = 1, bx = b415〉.
1. Hempel C. E. Metacyclic groups // Communs Algebra. – 2000. – 28. – P. 3865 – 3897.
2. Hyo-Seob Sim. Metacyclic groups of odd order // Proc. London Math. Soc. – 1994. – 69. – P. 47 – 71.
3. Bidwell J. N. S, Curran M. J. The automorphism group of a split metacyclic p-group // Arch. Math. – 2006. – 87. –
P. 488 – 497.
4. Curran M. J. The automorphism group of a split metacyclic 2-group // Arch. Math. – 2007. – 89. – P. 10 – 23.
5. Golasinski M., Goncalves D. L. On automorphisms of split metacyclic groups // Manuscr. Math. – 2009. – 128. –
P. 251 – 273.
6. Curran M. J. The automorphism group of a nonsplit metacyclic p-group // Arch. Math. – 2008. – 90. – P. 483 – 489.
Received 03.04.12
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 10
|
| id | umjimathkievua-article-2670 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:27:59Z |
| publishDate | 2012 |
| publisher | Institute of Mathematics, NAS of Ukraine |
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| resource_txt_mv | umjimathkievua/04/ed93cfdb5dcff83688edb72d968ddf04.pdf |
| spelling | umjimathkievua-article-26702020-03-18T19:32:22Z On split metacyclic groups Про суми Гаусса та узагальненi числа Бернуллi Liu, He-guo Yan, Yang Лю, Хе-го Ян, Янг Using the properties of primitive characters, Gauss sums, and the Ramanujan sum, we study two hybrid mean values of Gauss sums and generalized Bernoulli numbers and give two asymptotic formulas. Iз використанням примiтивних характерiв, сум Гаусса та суми Рамануджана вивчено два гiбридних середнiх значення сум Гаусса й узагальнених чисел Бернуллi та отримано двi асимптотичнi формули. Institute of Mathematics, NAS of Ukraine 2012-10-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2670 Ukrains’kyi Matematychnyi Zhurnal; Vol. 64 No. 10 (2012); 1432-1437 Український математичний журнал; Том 64 № 10 (2012); 1432-1437 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/2670/2099 https://umj.imath.kiev.ua/index.php/umj/article/view/2670/2100 Copyright (c) 2012 Liu He-guo; Yan Yang |
| spellingShingle | Liu, He-guo Yan, Yang Лю, Хе-го Ян, Янг On split metacyclic groups |
| title | On split metacyclic groups |
| title_alt | Про суми Гаусса та узагальненi числа Бернуллi |
| title_full | On split metacyclic groups |
| title_fullStr | On split metacyclic groups |
| title_full_unstemmed | On split metacyclic groups |
| title_short | On split metacyclic groups |
| title_sort | on split metacyclic groups |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2670 |
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